Acta of Bioengineering and Biomechanics Original paperVol. 17, No. 4, 2015 DOI: 10.5277/ABB-00128-2014-04
Skinfold creep under load of caliper.Linear visco- and poroelastic model simulations
JOANNA NOWAK1*, BARTOSZ NOWAK2, MARIUSZ KACZMAREK1
1 Institute of Mechanics and Applied Computer Science, Kazimierz Wielki University Bydgoszcz, Poland.2 Faculty of Civil Engineering, Bauhaus-Universität Weimar, Weimar, Germany.
Purpose: This paper addresses the diagnostic idea proposed in [11] to measure the parameter called rate of creep of axillary fold oftissue using modified Harpenden skinfold caliper in order to distinguish normal and edematous tissue. Our simulations are intended tohelp understanding the creep phenomenon and creep rate parameter as a sensitive indicator of edema existence. The parametric analysisshows the tissue behavior under the external load as well as its sensitivity to changes of crucial hydro-mechanical tissue parameters, e.g.,permeability or stiffness. Methods: The linear viscoelastic and poroelastic models of normal (single phase) and oedematous tissue (two-phase: swelled tissue with excess of interstitial fluid) implemented in COMSOL Multiphysics environment are used. Simulations areperformed within the range of small strains for a simplified fold geometry, material characterization and boundary conditions. The pre-dicted creep is the result of viscosity (viscoelastic model) or pore fluid displacement (poroelastic model) in tissue. Results: The tissuedeformations, interstitial fluid pressure as well as interstitial fluid velocity are discussed in parametric analysis with respect to elasticitymodulus, relaxation time or permeability of tissue. The creep rate determined within the models of tissue is compared and referred to thediagnostic idea in [11]. Conclusions: The results obtained from the two linear models of subcutaneous tissue indicate that the form ofcreep curve and the creep rate are sensitive to material parameters which characterize the tissue. However, the adopted modelling as-sumptions point to a limited applicability of the creep rate as the discriminant of oedema.
Key words: computer simulations, soft tissue, lymphoedema, viscoelasticity, poroelasticity, modeling
1. Introduction
A treatment of breast cancer causing disturbancein lymph transport from interstitial space throughlymphatic system to blood can result in accumulationof lymph in tissue of trunk and arm [10]. A simplemeasure used in evaluation of swelling of the poste-rior axillary tissue is based on the comparison ofthickness of the tissue’s fold on the normal and af-fected sides. The alternative measure, as proposed in[11], may constitute a comparison of change of thick-ness of the fold of such tissues determined at twocharacteristic time instants under approximately con-stant load exerted by modified Harpenden skinfoldcalliper. The authors of [11] determined the time in-stants (10 s and 60 s), the optimum loads (37 kPa) due
to the action of the calliper’s spring and made seriesof tests confirming differences in the behaviour ofnormal and affected tissue. The phenomenon which isconsidered as responsible for the thickness changes iscreep phenomenon of tissue. Since the tissue withedema contains more interstitial fluid than the normaltissue [8], the observed creep is usually larger in theformer case. While this original idea is very interest-ing for clinicians the literature confirming the possi-bility of axillary edema assessment throughout analy-sis of creep is very limited and this is a possiblereason that this simple methodology is not widelyused. The creep phenomenon of the lymphedematoustissue was also mentioned in papers [3] and [13],where another diagnostic method, called indentationtest or tonometry, was considered. The results indi-cated greater sensitivity of response of lymphedema-
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* Corresponding author: Joanna Nowak, Institute of Mechanics and Applied Computer Science, Kazimierz Wielki University,Kopernika 1, 85-074 Bydgoszcz, Poland. Tel: +48 252 325-76-53, e-mail: [email protected]
Received: July 7th, 2014Accepted for publication: January 12th, 2015
J. NOWAK et al.40
tous tissue to mechanical load than observed in thenormal tissue.
This paper presents results of simulations of thecreep of a skinfold assuming two mechanical models oftissue: viscoelastic and poroelastic one. Assuming thatnormal tissue can be modelled as a viscoelastic materialand the edematous tissue is a fluid saturated porousmaterial the models are considered in order to explorepossible behaviour of normal and edematous skinfold.The influence of material properties characterising thetissue on mechanical and hydro-mechanical processesunder constant stress imposed by calliper tips is ana-lysed. This parametric analysis may deliver betterknowledge on possible sources of lack of success of thecreep measure in diagnostic practice.
We assume simplified linear models of soft tissuewith homogeneous and isotropic material properties.This means that the presence of skin is not incorpo-rated. Additionally the fold geometry as well asboundary conditions are idealized and the symmetryof the process is assumed. The load level considered isalmost five times lower than the originally applied in[11] in order to limit the resultant strain which has tobe acceptable within geometrically linear models. Thecontrol time instants are the same as found in [11].
The discussion of results obtained by finite elementmethod is concentrated on tissue deformations, intersti-tial fluid pressure, and fluid velocity. The deformationof tissue under calliper tips represents the creep phe-nomenon due to internal friction or displacement ofinterstitial fluid. The analysis refers to control pointslocated beneath and next to the loading domain. Para-metric analysis reveals the role of elasticity modulus,viscosity of permeability of tissue for creep behaviour.
2. Materials and methods
In Fig. 1, a picture of a patient with truncal lym-phoedema tested with modified Harpenden skinfold
caliper and a simplified geometry of the skinfold whichis used in simulations are shown.
The soft tissue is modeled as the viscoelastic or po-roelastic material. Only in the latter case the presenceof interstitial fluid is represented explicitly by the porefluid which saturates porous skeleton. In both cases weassume negligible role of inertial and gravity forces aswell as the symmetry of stress tensor. Similar level ofcomplexity (number of model parameters) is postulatedwithin the constitutive mechanical relationships of themodels applied. The presence of skin, inhomogeneityand anisotropy of the material are disregarded.
2.1. Viscoelastic model
A single phase viscoelastic model of solid materialwhich incorporates the assumptions introduced isbased on the equilibrium equation
0σ =⋅∇ (1)
where σ is the stress tensor. Following the concept ofthe generalized Maxwell model of the viscoelasticcontinuum with internal variables [2], the stress tensoris decomposed into volumetric stress (pressure) P andstress deviator s
sIσ +−= P (2)
and then the constitutive relationships for the linearviscoelastic material are written as the equations
,1
,22,P1
dtd
dtd
GGK
dj
j
j
n
jjjdv
εqq
qεs
=+
+=−= ∑=
τ
ε
(3)
where the strain tensor ε defined by the displacementvector u
is decomposed into volumetric strain εv = trε
and tensor of strain deviator εd as follows ε = 21 [(∇u)
Fig. 1. A test with the modified Harpenden caliper [9] (a) and simplified geometry of the skinfold used in simulations (b)
a) b)
Skinfold creep under load of caliper. Linear visco- and poroelastic model simulations 41
+ (∇u)T] = 31 εvI + εd; K =
)3(3 EGEG−
is the bulk
modulus, G is the shear modulus, E stands for Young’smodulus, Gj represent stiffness of arms ( j = 1, ..., n)in the generalized Maxwell model, q j denote thesymmetric tensors of internal state (internal strains)and τj are relaxation times [2]. In the applied model oftissue the simplest approximation is considered, i.e.,i = 1, which guarantees similar complexity of the vis-coelastic and poroelastic models.
2.2. Poroelastic model
The poroelastic model of soft tissue is based hereon Biot’s formulation [4] and comprises the equilib-rium equation of the two phase system and Darcy’slaw, i.e.,
0σ =⋅∇ , (4)
p∇−=ηkq , (5)
where now σ denotes the total stress (sum of the stressin solid and fluid phase), q is the discharge velocity ofpore fluid with respect to solid skeleton, p stands forpore pressure, k and η denote permeability of porousmedium and dynamic viscosity of fluid.
The linear constitutive relationships for mechani-cal behaviour of fluid saturated porous material arethe following
IεΙσ vλεμα +=+ 2p , (6)
vpM
αεζ +=1 , (7)
where ζ denotes the change of fluid content in porousmatrix, μ = G and λ are Lame elastic constants ofdrained porous matrix describing response of solidskeleton in terms of the effective stress to the appliedstrain, α is the volumetric coupling coefficient, andM is the elasticity constant relating the change of fluidcontent and pore pressure, [4].
From equilibrium equation (4) and constitutiverelation (6) we have
0)()2(2 =∇−∇⋅∇++∇ pαλμη uu . (8)
Since the change in fluid content ζ can be ex-pressed by the product of porosity φ and difference ofdilatations of pore fluid θ and solid matrix εv, i.e.,
)( vεθφζ −= , (9)
and the linearized dependence of the divergence ofdischarge velocity can be represented as
)( vtεθφ −
∂∂
=⋅∇ q , (10)
combining equations (5), (7) and (10) we can write
01=
∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛∇−⋅∇+
∂∂
vtpk
tp
Mεα
η. (12)
Expressing the Lame constants by Young’smodulus E and Poisson’s ratio ν of drained material
μ = ,)1(2 ν+
E λ = )21)(1(2 νν
ν−+
E and representing
parameter M for incompressible matrix material as
M = ,φ
fK [4], where Kf denotes the fluid compressi-
bility, we get the following coupled system of equa-tions of the poroelastic model
)0)()21)(1(2)1(2
2 =∇−∇⋅∇−+
+∇+
pEE αννν
uu (13)
0=⋅∇∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∇−⋅∇+
∂∂ u
tpk
tp
K fα
ηφ . (14)
Equations (13), (14) will be used to model the be-havior of subcutaneous tissue as fluid saturated porousmaterial in terms of matrix displacement u and porepressure p.
2.3. Model parameters,initial and boundary conditions
Table 1 lists the parameters characterizing the skin-fold geometry and material properties of the tissueassumed for simulations.
Due to the symmetry of the mechanical problem ofthe loaded skinfold only half of the fold, shown in Fig.1b, with the adjacent tissue is taken into account. Thegeometry of this part of the fold and the cross-sectionare shown in Fig. 2. The length, width and half of thethickness of the simulated fold are L = 8 cm, Wf =6 cm and d = 2 cm while the width and thickness ofthe adjacent tissue are Wt = 6 cm and D = 2 cm. Therectangular coordinate system is used with x axis ori-ented along the fold. The load pc = 7.8 kPa is appliedto the surface of contact of skin with the caliper tip.The geometrical parameters for the caliper tips areselected following paper [11]. The size of the tip ischaracterized by length l = 15 mm and width w =6 mm. Then, the surface area of the contact is equal to
J. NOWAK et al.42
90 mm2. The assumed load on caliper tips is aboutfive times lower than the value considered in [11].The values of basic material parameters characterizingmechanical and hydraulic properties of soft tissue areadopted from literature, see [1], [6]–[8], [12] and werediscussed in [7]. The parametric analysis is performedfor a range of model parameters being deviations fromtheir basic values and ensuring that results of simula-tions satisfy the accepted limitations. Studies of theporoelastic model with respect to variation of stiffnessof tissue assume three values of drained Young’smodulus E = 2/2.5/3 × 104 kPa and the same for eachcase drained Poisson’s ratio ν = 0.33, while the pos-sible range of hydraulic properties of tissue is ex-pressed by the three values of permeability which are
k = 0.5/1.5/4.5 × 10–13 m2. The studies of the vis-coelastic model assume elastic parameters (E, G) thesame as the drained elastic parameters of the poro-elastic model. The other two coefficients of the Max-well model (G1,τ) are selected to ensure that the pre-dictions for initial deformation level and rate of creepare comparable for the viscoelastic and the poroelasticmodels. As a result, the parameter of stiffness of anarm in Maxwell model G1 = 3.6 kPa, and the values ofrelaxation time τ = 3/7/16 s.
Initially the pore fluid pressure and tissue displace-ment are assumed to be equal to zero. The relative fluidflow is not allowed through skin and symmetry planes.The displacements of tissue for boundaries which areincluded by symmetry planes are equal to zero.
Table 1. Geometrical and material parameters assumed for simulations
Parameter Description Value UnitsL Length of fold 8.0 cmWf Width of fold 6.0 cmWt Width of adjacent tissue layer 6.0 cmd Thickness of half of the fold 2.0 cmD Thickness of tissue layer 2.0 cma Depth of control points 0.05 cmb Distance between control points 1.0 cml Length of caliper tip 1.5 cmw Width of caliper tip 0.6 cmα Volumetric coupling coefficient 1.0 –ρ Liquid density 1000.0 kg/m3
ν Poisson’s number of the tissue 0.33 –E Young’s modulus of the tissue 2/2.5/3 × 104 Paφ Porosity 0.05 –Kf Liquid compressibility 2.3 × 109 Pak Hydraulic permeability 0.5/1.5/4.5 × 10–13 m2
η Dynamic viscosity 0.001 Paspc Mechanical load (normal stress) 7.78 (58, 32) kPa (mmHg)G1 Parameter of stiffness in Maxwell model 3.6 kPaτ1 Relaxation time in Maxwell model 3/7/16 s
Fig. 2. Geometry of half of the fold (a) and cross-section through the element (b)
a) b)
Skinfold creep under load of caliper. Linear visco- and poroelastic model simulations 43
3. Results
The results of numerical simulations of the skin-fold behavior under load of caliper using the aboveformulated models and the finite element methodare implemented in COMSOL Multiphysics envi-ronment is presented in the next few figures. Twocontrol points G and H, located in different parts ofthe fold (see Fig. 2b), are selected in order to studythe skinfold behavior. The points are placed 0.5 mm(in Fig. 2b distance a) beneath the surface of the fold.Point G is located just below the center of the calipertip while point H is placed next to the tip, 1 cm (dis-tance b) from point G (see Fig. 2b).
(a) Parametric analysis for 3D geometry
The evolution (time dependence) of tissue dis-placement, pore pressure and pore fluid velocity forselected model parameters (given in Table 1) arestudied. As in paper [11], the change of thickness ofthe fold at two time instants (10 s and 60 s) are com-pared to estimate the creep rate parameters. Figures 3,
4 and 5 show displacements of tissue at control points Gand H as functions of time predicted by the viscoelas-tic and poroelastic models. The level of deformationat the point beneath the caliper tips (G) for all thecases considered is few times larger than for the tissuenext to the tips (H).
The displacements under the tip (point G) dependvisibly on Young’s modulus while next to the tip(point H) the dependence is less pronounced, see Fig. 3.The slope of creep curves obtained in the case of theviscoelastic model (Fig. 3a) is similar to that obtainedfrom the poroelastic model (Fig. 3b).
The plots in Fig. 4 visualize the dependence of tis-sue displacement on factors that influence the fold’screep: permeability and relaxation time. The form oftime dependence of displacement predicted by bothmodels at the same control points are different al-though the levels of values are comparable. The dis-placements under the tip are about five times greaterthan next to the tip.
The results obtained within the viscoelastic modelshow that the rate of deformation in the observationwindow is greater for lower values of relaxation time.Given the values of relaxation time 3, 7, and 16 s the
Fig. 3. Dependence of tissue displacement on time at control points G and H for the viscoelastic model (a)and the poroelastic model (b) and selected values of Young’s modulus
Fig. 4. Dependence of tissue displacement on time at control points G and H for the viscoelastic model (a)and the poroelastic model (b) assuming different values of relaxation time and permeability
a) b)
a) b)
J. NOWAK et al.44
creep rate parameter as defined in [11] amounts to0.034, 0.125, and 0.187 mm, respectively. The de-formation process reaches equilibrium after about 22,42 and 86 s.
Fig. 5. Comparison of tissue displacement as a function of timeat control point G for the viscoelastic and poroelastic modelsassuming different values of relaxation time and permeability
For the poroelastic model the rate of deforma-tion increases with the values of permeability. Thevalues of the creep rate for permeability 5, 15, and45 10–14 mm2 are 0.117, 0.094, and 0.0062 mm, re-spectively, and for all the cases the deformation doesnot stabilize within the range of observation (60 s). Atpoint H the sensitivity of deformation with respect torelaxation time is lower than with respect to perme-ability. The effect is related to outflow of fluid fromthe loaded domain to its vicinity. Higher permeabilitycauses that interstitial fluid flows out faster from thetissue, therefore the material can deform easier andthe time which is necessary to reach the equilibriumstate is shorter.
For better visualization of the difference betweencreep of skinfold predicted by the two models consid-ered, displacements at point G are compared in Fig. 5.The adopted values of parameters of stiffness ensurecomparable initial displacement of tissue for the vis-coelastic and the poroelastic models. The permeabilityand the relaxation time influence the form of the creepcurves and duration of the process. The initial highrate of deformation of the fold is comparable for both
Fig. 6. Dependence of pore fluid pressure (a) and velocity (b)on time at points G and H for different Young’s moduli
Fig. 7. Dependence of pore pressure (a) and velocity (b)on time for different values of permeability at control points G and H
a) b)
a) b)
Skinfold creep under load of caliper. Linear visco- and poroelastic model simulations 45
models. Then, the rate predicted by the poroelasticmodel slows down compared to the one obtained fromthe viscoelastic model. The observed behavior can beexplained by the fact that initially the pore fluid isrelatively fast squeezed out from the tissue beneaththe tips to the unloaded domain surrounding tips be-cause the pressure gradient is high. In the second pe-riod the pore pressure gradient becomes more homo-geneous (this is visible in Fig. 8a) and lower while therate of fluid outflow from the loaded tissue slowsdown.
The results shown in Fig. 6 and 7 concern para-metric studies of the poroelastic model for changes inpore pressure and fluid velocity under caliper tip(point G) and next to the tip (point H) as a function oftime. Different values of Young’s modulus (E = 2,2.5, and 3 × 104 Pa) and identical permeability (k =1.5 × 10–13 m2), Fig. 6, or constant Young’s modulus(E = 2.5 × 104 Pa) and different values of permeability(k = 5, 15 and 45 × 10–14 m2), Fig. 7, are considered.
The results show significant difference of pressurevalues at control points in the initial period (approx.10 s), Fig. 6a. Because of the possibility of expansionof tissue next to the caliper tips the pressure in thispart is much lower than under the tips, although thedistance is only 1 cm. The same effect induces that thelower the Young’s modulus is, the lower the pressureis observed. Within the observation time range con-sidered the pressure dissipates almost to zero.
The changes in pore fluid velocity at control points,see Fig. 6b, show weak dependence on Young’s
modulus. Moreover, it is seen that the velocity ap-proaches fast the values close to zero. The pore fluidvelocity in tissue next to the tip (point H) has longernon-zero values and is more dependent on Young’smodulus of tissue than the fluid velocity at point G.
The results shown in Fig. 7a and 7b refer to para-metric studies of influence of permeability on porepressure and pore fluid velocity in tissue under calipertip (point G) and next to the tip (point H).
The lower the permeability, the higher the porepressure is present both under the tip (point G) and inthe vicinity (point H). This behavior is physically welljustified because of easier fluid outflow in the cases ofhigher permeability. Only for the highest permeabilityfull dissipation of pore pressure is possible within 60 s.The above explanation is supported by the dependenceof pore fluid velocity on tissue permeability, Fig. 7b, andthe fact that the flow becomes relatively slow for lowpermeability tissue. The velocity is particularly highin the initial time period and at point H is significantlyhigher than at point G.
b) Spatial distributions for 2D geometry,comparison with the 3D caseThe longitudinal shape of the skinfold and caliper
tips is justified in spite of the simulations for 3D casesperforming calculations for 2D geometry. Then, theinfinite size of the fold and tips is assumed and simu-lations are noticeably less time consuming. The com-plexity of the solution for 3D geometry had significantinfluence on the number of mesh elements and the
Fig. 8. Spatial distributions of pore pressure (a) and fluid velocity (b) for loaded skinfold (7.78 kPa)and two time instants 10 and 60 s (poroelastic model)
J. NOWAK et al.46
calculation time. Typically, in the 2D case the num-ber of mesh elements was three times lower and thecalculation time ten times shorter than for the 3Dcase.
Spatial distributions of pore pressure and porefluid velocity obtained from 2D poroelastic model fortime instants 10 and 60 s are shown in Fig. 8.
The evolution of pore pressure and fluid velocityat control points G and H derived from 2D and 3Dcases are compared in Fig. 9. The results showinghigh pore pressure for 10 s in the loaded area of tissueindicate that initially the pore fluid takes on most of
the load from caliper tips. The pressure is particularlyhigh under and in the vicinity of tips. With time thepressure goes down and for 60 s it is approximatelythree times smaller than the maximum value. Theprocess is related to evacuation of pore fluid fromloaded domain, which is visible on plots presentingpore fluid velocity distributions (Fig. 8a). The porepressure next to the tips (point H) is initially low andincreases up to values comparable with pressure atpoint G, see Fig. 9a. The process could be explainedby redistribution of pore fluid and limited deformationof tissue matrix next to the tips.
Fig. 9. Comparison of pore pressure (a) and fluid velocity (b) at control points.The results are obtained from 2D and 3D simulations within the poroelastic model
Fig. 10. Spatial distributions of tissue displacement from the viscoelastic (a)and the poroelastic (b) models for time instants 10 and 60 s
a) b)
Skinfold creep under load of caliper. Linear visco- and poroelastic model simulations 47
In Fig. 10, the distributions of tissue displacementsin direction z obtained for 10 and 60 s from the vis-coelastic and the poroelastic 2D simulations are illus-trated. Then, in Fig. 11, the changes in displacementsversus time at control points obtained for 2D and 3Dgeometry are compared.
Large displacements are observed particularly inthe vicinity of the loaded region. The spatial changesin displacement both for 10 and 60 s are less signifi-cant for the viscoelastic model than for the poroelas-tic model (Fig. 10). Despite the same elastic proper-ties the model predictions for time evolution ofdisplacement for 2D and 3D geometry visibly diverge(Fig. 11). The displacements at points H and G differby one order of magnitude.
4. Discussion
Studies of deformation of skinfold under loadingof caliper tips (Fig. 3–5) have shown that the shape ofcreep curves and the creep measure assumed in [11]which is the difference between thickness of loadedtissue at 60 s and 10 s are sensitive to material prop-erties of the viscoelastic and poroelastic models. Theform and rate of creep curves obtained from bothmodels are similar when changing the stiffness, Fig. 3.However, when changing the permeability and re-laxation time (factors responsible for creep) the formof time dependence of displacement becomes differ-ent, Fig. 4. The mechanisms of internal viscous fric-tion of solid (included in the viscoelastic model) andviscous interaction of pore fluid with porous solidskeleton (included in the poroelastic model) causevisibly different time behavior of deformation of theskinfold. While the clinical results obtained in [11]have shown that the creep rate can be a quantitative
measure being descriptor of tissue’s state, more sensi-tive than the skinfold thickness, the results of oursimulations can confirm this behavior only in the caseof short relaxation time, Fig. 5. When the relaxationtime of the viscoelastic model of normal tissueamounts to 3 s or less the creep rate predicted by the
poroelastic model of tissue, referred to its edematousstate, is significantly larger in comparison with thevalues obtained from the viscoelastic model. Whenthe relaxation time is longer the creep rate appears tobe not a good indicator of edema. This may justify thefact of limited application of the diagnostic idea pro-posed in [11] in clinical practice. The doubts have alsofound confirmation in results of relatively large set oftests performed in [9]. It should be noticed that thepredicted differences of the forms of dependence ofdeformation of skinfold on time (compare results inFig. 5) are not fully represented by the creep rate pa-rameter considered and another more appropriate de-scriptor should be searched to express the discrepan-cies.
Although the geometry of skinfold and caliper’stips may suggest that 2D geometry of the problemcould be a good approximation the results of simula-tions have proven significant quantitative differencesbetween predictions for 2D and 3D cases. The resultsfor 2D geometry, however, are useful to explain de-tailed physical phenomena associated with the macro-scopic response of the skinfold.
The above results and features of the models con-sidered are significant from diagnostic viewpoint if themodels and contributing mechanisms are well recog-nized and the range of material parameters properlyselected for the real human tissues in normal and ede-matous state. Although in the literature (see the reviewpaper by Wiig and Swartz [14]) the number of factorscharacterizing complex structure of the tissues are dis-cussed, e.g., interstitial fluid content, structure of colla-
Fig. 11. Comparison of tissue displacement at points G (a) and H (b)for the poroelastic and the viscoelastic models
a) b)
J. NOWAK et al.48
gen and elastin fibers, structure of skin, presence orabsence of the lymphatic vessels, etc., up to date de-tailed mechanisms responsible for the observed behav-ior of mechanically loaded tissues (including creep) arenot fully understood and corresponding macroscopicmodels not validated. Due to the deficiency of data onhuman lymphedematous tissue in different pathophysi-ological stages, such as stiffness or hydraulic conduc-tivity, the possibilities of developing reliable simula-tions of the tissue behavior are limited.
The results of simulations performed cannot becompared directly to the available clinical data be-cause due to limitation used to apply the linear modelsthe loads in our simulations were almost five timeslower than assumed in studies in vivo. Taking intoaccount the advantages of linear modeling (validity ofsuperposition principle) the appropriate methodologyof testing in vivo with much lower loading levelshould be considered as the alternative to use the cur-rently accepted loads and models within large defor-mation range.
5. Conclusions
The simulations of skinfold under the load ofmodified Harpenden caliper were elaborated. Thepredictions of the linear viscoelastic and poroelasticmodels were compared for 3D and 2D geometry. Thetime evolution and spatial distributions of tissue dis-placement, pore fluid pressure and pore fluid velocitywere analyzed.
The results of parametric studies show that the formof creep curves and the creep rate predicted by themodels considered depend on tissue’s stiffness andfactors responsible for internal friction (relaxation timeor permeability). In the light of the simulations thecreep rate parameter defined in [11] appeared to havelimited value as the indicator of edema. This is onlypossible in cases when normal tissues described withinthe viscoelastic model have very short relaxation time.The viscoelastic model does not deliver explanation ofspecific creep behavior while the same effects seen interms of the poroelastic model of tissue can be betterphysically understood taking into account the pore fluiddisplacement and pressure dissipation. The discrepan-cies of predictions for 2D and 3D models are signifi-cant and this means that the former one can be onlya rough approximation of the process considered.
The understanding of the hydro-mechanical be-havior of edematous tissue of skinfold under load canbe used in the development of testing methodology
and measuring devices. The size and shape of cali-per’s tips, the time of observations and other parame-ters could be optimized. Further modeling studies ofthe skinfold creep should incorporate large deforma-tion range and loads corresponding to the values ap-plied in diagnostic tests.
Acknowledgements
This work was partially supported by the National ScienceCentre in Poland under grant 2011/01/B/ST8/07283.
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